When evaluating double integral using polar co-ordinates, does the order of $dr ~ d\theta$ make any difference?
Suppose,
$$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$ $$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 d\theta dr$$
Do the above question yield different answers? If the process for solving the above is different can you please explain in general?
Sorry I have no formatting knowledge, so any light here would be helpful for any future questions.
The first integral makes perfect sense; the second one is nonsensical.
I write my integrals with the $d$ first to eliminate any confusion as to what variable goes with what integral:
$$\int_0^{\pi/4} d\theta \: \int_0^{\sin{\theta}} dr \,r^2$$
Evaluate right to left. Clearly, $r$ depends on $\theta$, and integrate over $\theta$. If you wish to switch the order of integration, you must, as Mhenni points out, redefine your integration region:
$$\int_0^{1/\sqrt{2}} dr \, r^2 \int_{\arcsin{r}}^{\pi/4} d\theta$$